Exploring a Data Set: Plotting with Error Bars

The Data Set

At the URL http://pdg.lbl.gov/~sbl/pipp_total.dat
there is a data set that describes the total interaction cross section
for the elementary particle reaction .
After downloading the file (named pipp_total.dat)
from the web we can read it into Mathematica using the Import
function. (If the file pipp_total.dat
is not placed in a directory that is contained in Mathematica's
$Path variable,
you can make it available by executing the function
where path is the path to the needed directory.)

The first 11 lines of this data set consists of an informational
header, the remaining lines contain data. The fifth row names the
data that are in each column (there are 11 columns). The columns
in the data set are as follows:

From this we see that, column 2 contains the laboratory beam energy,
column 5 contains the cross section, and columns 6 and 7 contain
the errors. First we drop the header information from the data

And now, using pattern matching and transformation rules, we create
data sets with the desired information. The list dataWithErrors
contains the data along with the error information in the form
and the list dataWithoutErrors
contains just the
data.

Plotting the data

First we make a basic plot of the data.

There are, in fact, data beyond 14 GeV/c laboratory energy.

This data could be seen by choosing the
option to ListPlot.
However, to exhibit the structure of the data, a log-log plot would
serve better. Various logarithmic plots are contained in the standard
add-on package Graphics`Graphics`.
First load the package.

Now take a look at the data again.

Select the data between 1 and 2.5 GeV/c to get a closer look in
that region.

Plotting with Error Bars: MultipleListPlot

One way that we can put errors onto this plot is to make use of
the standard add-on package Graphics`MultipleListPlot`.

There are several ways to specify the points
in the data sets used by MultipleListPlot.
The most basic involve only the locations of the points. If
a point is given as a single number, the value is treated as
a
value, and the
value is determined by the position of the point in the list
of data. If given as a pair ,
the point is placed at the
coordinates. Each point can be grouped with an error specification,
given as an ErrorBar
object. An ErrorBar
object can accept a specification of error in the
variable or in both
and .
An error given as a single value indicates the same error in
the positive and negative directions; a negative number paired
with a positive number indicates a different error in each direction.

First load the package.

Loading this package then enables the ErrorBar
function.

Here is the usage message for ErrorBar:

The selected data can be put into this form through a transformation
rule.

It appears from looking at this plot that, of the various data
sources that were combined to produce this overall data set, there
appear to be some (perhaps one?) that have statistics that are quite
different from the others. One could continue to investigate this
by partitioning the data according to the "Reference"
field to discover whether this hypothesis is true.

Plotting with Errors: Graphics Primitives

Lets take a different approach to producing this plot. Here we
will work with graphics primitives to gain more control over the
form that this plot takes (although Graphics`MultipleListPlot`
does give you more control than we have used above), and to ultimately
produce a customized graphic.

Select the data between 1 and 2.5 GeV/c again

Our approach will be to convert the data of the form
into graphics directives: Points
and Lines. Thus,
for each data point to have error lines extending above and below,
we would want something of the form .
We can do this easily with a transformation rule.

Display the graphics primitives by enclosing them in Graphics
and then use Show.

Let's color the error bars red and make the data points larger.

Now we color the error bars blue when either of the errors is
greater than 0.5

This sort of graphics development and exploration can be continued
along these lines so as to customize a set of data visualization
tools for your specific needs. (A simple extension of this example
would be to add the serifs on the tops and bottoms of the error
bars: this just involves the addition of some more line primitives
to the transformation rules used here.) You could then gather the
resulting code together to produce a function that can be reused
as needed. Here is a basic example that makes use of what we have
done here.